Hamiltonian formalism for hard and soft excitations in a plasma with a non-Abelian interaction

Hamiltonian theory for collective longitudinally polarized gluon excitations (plasmons) interacting with classical high-energy color-charged test particle propagating through a high-temperature gluon plasma is developed. A generalization of the Lie–Poisson bracket to the case of a continuous medium involving bosonic normal field variable $a^{a}_{\boldsymbol{k}}$ and a non-Abelian color charge $Q^{a}$ is performed and the corresponding Hamilton equations are derived. The canonical transformations including simultaneously both bosonic degrees of freedom of the soft collective excitations in the hot gluon plasma and the degree of freedom of a hard test particle associated with its color charge are presented. A complete system of the canonicity conditions for these transformations is obtained. An explicit form of the effective fourth-order Hamiltonian describing the elastic scattering of a plasmon off a hard color particle is found and the self-consistent system of Boltzmann-type kinetic equations taking into account the time evolution of the mean value of the color charge of this particle is obtained.